On the Bernstein's constant in convex approximation

TL;DR

This paper investigates the asymptotic behavior of the best uniform convex polynomial approximation of the absolute value function raised to a power, focusing on the existence of a specific limit related to Bernstein's constant.

Contribution

It addresses an open question about the existence of a limit involving convex polynomial approximation of |x|^λ, contributing to the understanding of Bernstein's constant in this context.

Findings

Open question remains unresolved about the limit's existence.

Provides new insights into convex approximation of |x|^λ.

Highlights challenges in asymptotic analysis of convex polynomial approximation.

Abstract

Denoting by $E_{n}^{(+2)}(f)$ the best uniform approximation of $f$ by convex polynomials of degree $\le n$, there is an open question if there exists the limit $\lim_{n\to \infty}n^{\lambda}E_{n}^{(+2)}(|x|^{\lambda})$ for $\lambda \ge 1$.